Huh, how comes they claim "We believe in privacy" and then use a closed-source product instead of the Apache-licensed Dalvik?

"There are no back doors or anything third parties could use for monitoring your activity. Together with the open source community, we’re continuing to strengthen our privacy capabilities at every opportunity."

It would be interesting to know, as you say, why they don't just use Dalvik directly.

How do you install apps on it, can you get the Play Store?

Ever seen this: http://apps.evozi.com/apk-downloader/ ?

Officially, you are not able/allowed to install the Play Store. In practice, it can be installed easily and it works fine.

You can use other App stores like the Amazon app store or Aptoid.